Assign material to multiple objects in edit mode
How to assign materials to all or part of a single object
This is a simple process. Here's a screenshot:

Go into edit mode. Select the geometry that you want to be a different material (you can select faces individually, or press L when hovering your mouse over a mesh). Then simply click on Assign as shown in the screenshot above.
How to assign materials to multiple objects while in Multi-Object editing mode
Select one of your objects and create two material slots, and then also create two materials. Tweak your material settings for the second material to whatever you wish. I just used a bright blue color here.

Select all of the objects that you want to edit, shift click the first one so it becomes active.

Press CTRL + L to open the links pop-up, and then select Materials.

Make sure the number that appears on the materials panel matches the number of objects you have.

Select all of your objects and TAB into edit mode. Select what you want to change, be it faces or meshes, and then simply assign the new material.

Done!